Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C83 ⋊ #408 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D83 |
| commutator, socle | C22×C83 | C22×C4 |
| Fitting | C2×#41×C83 | C2 |
| hypercenter | C2×#41 | D83 |
| hypocenter | C83 | #408 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
158 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=e,e^2=f,b^a=bd,g^a=g^82,c^b=cf848 conjugacy classes — view the full character table.
| maximal subgroup of | #24907412, #24907414, #24907427, … |
| maximal normal subgroup of | #24907412, #24907414, #24907427, … |
| maximal quotient of | #24907160, #24907229, #24907375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 516);